C481 B581 Computer Graphics
Dana Vrajitoru

2D Transformations

Definition. A function F that maps each point (x0, y0) from an original area to a point (x, y) = F(x0, y0) in the transformed area. In general T is a function with 2 components, (Fx, Fy).

Direct computation:
for each (x0, y0) {
  getPixel(x0, y0, color);
  (x, y) = F(x0, y0);
  setPixel(x, y, color);
}

Drawback: although transformations are continuous functions in the real plane, when applied to a discrete area of pixels,  they may create gaps in the transformed area because of the scan conversion.

General method: apply the inverse transformation.

determine the boundary of the transformed area by a direct transformation;
for each (x, y) {
  (x0, y0) = F-1(x, y);
  getPixel(x0, y0, color);
  setPixel(x, y, color);
}

By this method, we leave no pixel out of the transformed area.


Translation of vector T(xt, yt):
Direct transformation
(
x
y
) = (
xo
yo
 ) + (
xt
yt		 )
Inverse transformation
(
xo
yo
) = (
x
y
 ) - (
xt
yt		 )


Rotation about the origin with an angle a:

Rotation matrix:

Ra
=(
cos a
sin a
 - sin a
 cos a		 ) 

Direct transformation
(
x
y
) =
 Ra * (
xo
yo		 )
Inverse transformation
(
xo
yo
) =
 R -a * (
x
y
 )

 
(
xo
yo
) = (
cos (-a)
sin (-a)
 - sin (-a)
 cos (-a)		 ) * (
x
y		 )
(
xo
yo
) = (
cos a
- sin a
 sin a
 cos a		 ) * (
x
y		 )


Scaling of factors a and b:

a = 1.7,   b = 0.8
Direct transformation
(
x
y
) = (
 a   0
0   b 		) * (
xo
yo
 )
Inverse transformation
(
xo
yo
) = (
 1/a   0
0     1/b 		) * (
x
y
 )


Symmetry (reflection) with respect to an axis:

Symmetry with respect to Ox
Flip vertically (axis = Ox)
 (
xo
yo
) = (
 1   0
0   -1 		) * (
x
y
 )

 
Flip horizontally (axis = Oy)
 (
xo
yo
) = (
 -1   0
0    1 		) * (
x
y
 )

Fractals

Def. Shapes that show similar features at different sizes.

From the programmer's point of view: recursively defined drawings.

Mandelbrot set:

Examples:
 
 
The Koch curve:
 The Sierpinski triangle

The Mandelbrot Set The Julia Set